Since M. O. Kramer reported successful experimental results in 1957, there have been repeated attempts to reduce frictional drag in turbulent fluid flow over a surface by applying a passive compliant coating. Experimental results in this area have been mixed. Most investigators have reported a drag increase, while only a few have claimed drag reduction for turbulent flow. A number of theoretical studies have characterized the stability of the laminar boundary layer over a deforming surface and other studies have characterized the reaction of a coating to a fluctuating load. However, no rigorous analytical technique has been previously reported that has been used to successfully design a drag-reducing coating for turbulent flow.
In the past, passive coatings were tested without specification and full characterization of critical physical parameters, such as the frequency dependent complex shear modulus, density, and thickness. In order to achieve and ensure drag reduction with a viscoelastic coating, a methodology is required for selecting appropriate material properties and for estimating anticipated drag reduction as a function of configuration and velocity.
Relevant background information for associated technical topics is available in the literature, and may be useful due to the technical complexity of this invention. A classical discussion of boundary layer theory, including formulation of Navier-Stokes and turbulent boundary layer equations, is provided in Boundary-Layer Theory, by Dr. Hermann Schlichting, published by McGraw Hill, New York, seventh edition, 1979. A discussion of structures and scales in turbulent flows can be found in Turbulence, 1975, McGraw Hill, written by J. O. Hinze, and in xe2x80x9cCoherent Motions in the Turbulent Boundary Layer,xe2x80x9d in Annual Review of Fluid Mechanics, 1991, volume 23, pp. 601-39, written by Steven K. Robinson. Background on Reynolds stress types of turbulence models is found in the chapter, xe2x80x9cTurbulent Flows: Model Equations and Solution Methodology,xe2x80x9d written by Tom Gatski, and included in the Handbook of Computational Fluid Mechanics, published by Academic Press in 1996. Equations in fluid and solid mechanics are often expressed in indicial, or tensor, notation, for compactness. Chapter 2 in the text A First Course in Continuum Mechanics, by Y. C. Fung, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977, provides a brief introduction into tensor notation for mechanics equations. An introduction to finite difference methods, which are used to solve the system of momentum and continuity equations for a turbulent fluid, is provided in the text, Computational Fluid Dynamics for Engineers, written by Klaus Hoffman, and published in 1989 by the Engineering Education System i Austin, Texas. Descriptions of measured and mathematically modeled physical properties of polymers are found in the text, Viscoelastic Properties of Polymers by J. D. Ferry, Wiley, New York, 1980, 3rd edition. The article, xe2x80x9cLoss Factor Height and Width Limits for Polymer Relaxation,xe2x80x9d by Bruce Hartmann, Gilbert Lee, and John Lee, in the Journal of the Acoustical Society of America Vol. 95, No. 1, January 1994, discusses mathematical characterization of shear moduli for real viscoelastic, polymeric materials, including those approximated by the Havriliak-Negami approach.
Recently in the international literature (K. S. Choi, X. Yang, B. R. Clayton, E. J. Glover, M. Atlar, B. N. Semonev, and V. M. Kulik, xe2x80x9cTurbulent Drag Reduction Using Compliant Surfaces,xe2x80x9d Proceedings of the Royal Society of London, A (1997) 453, pp. 2229-2240). Choi et al. reported experimental measurements of up to 7% turbulent friction drag reduction for an axisymmetric body coated with a viscoelastic material. These experiments were performed in the United Kingdom, using coatings designed and fabricated in Russia at the Institute of Thermophysics, Russian Academy of Sciences, Novosibirsk, by a team headed by B .N. Semenov. The basic design approach was outlined in xe2x80x9cOn Conditions of Modelling and Choice of Viscoelastic Coatings for Drag Reduction,xe2x80x9d in Recent Developments in Turbulence Management, K. S. Choi, ed., 1991, pp. 241-262, Dordrecht, Kluwer Publishers. The Novosibirsk design approach is semi-empirical in nature, and does not take into account the full characterization of the complex shear modulus of the viscoelastic material, namely, the relaxation time of the material. The Novosibirsk design approach does take into account frequency-dependent material properties. Furthermore, the Novosibirsk concept is valid only for a membrane-type coating, such as a film which coats a foam-rubber saturated with water or glycerine, and where only normal fluctuations of the surface are considered.
The structure of coatings intended for drag reduction has been addressed in the international literature, starting with the 1938 patent No. 669-897, xe2x80x9cAn Apparatus for the Reduction of Friction Drag,xe2x80x9d issued in Germany to Max O. Kramer. Kramer later received a patent in 1964, U.S. Pat. No. 3,161,385, and in 1971, U.S. Pat. No. 3,585,953 for coatings to extend laminar flow in a boundary layer. Soviet inventor""s certificates, such as xe2x80x9cA Damping Covering,xe2x80x9d USSR patent 1413286, Publication 20.01.1974, Bulletin of the Inventions 14, by V. V. Babenko, L. F. Kozlov, and S. V. Pershin, xe2x80x9cAn Adjustable Damping Covering,xe2x80x9d USSR patent 1597866, Publication 15.03.1978, Bulletin of the Inventions 110, by V. V. Babenko, L. F. Kozlov, and V. I. Korobov, and xe2x80x9cA Damping Covering for Solid Bodies,xe2x80x9d USSR patent 1802672, Publication 07.02.1981, Bulletin of the Inventions 15, by V. V. Babenko and N. F. Yurchenko, have also described the structure of drag-reducing coatings comprised of viscoelastic materials. These inventor""s certificates identified the three-dimensional structure within a drag-reducing coating, but do not address the methodology for choosing appropriate parameters of the viscoelastic materials to be used in the manufacture of such coatings. Structural features include multiple layers of materials, longitudinal, rib-like inclusions of elastic, viscoelastic, or fluid materials, and heated elements. Viscoelastic coatings may be combined with other forms of structure, such as longitudinal riblets molded on or within the surface of the coating. As described in the international literature in publications such as xe2x80x9cSecondary Flow Induced by Riblets,xe2x80x9d written by D. B. Goldstein and T. C. Tuan, and published in the Journal of Fluid Mechanics, volume 363, May 25, 1998, pp. 115-152, two-dimensional, rigid riblets alone have been shown experimentally to reduce surface friction drag up to about 10%.
The present invention enables the design of a passive viscoelastic coating for the reduction of turbulent friction drag. Coatings with material properties designed using the methodology described in this invention have reduced friction drag by greater than 10%. The methodology of the present invention permits, as a first object of the invention, the specification of the frequency dependent complex shear modulus, the density, and the thickness of an isotropic viscoelastic material which will reduce turbulent friction drag relative to specific flow conditions over a rigid surface. Quantitative levels of drag reduction can be estimated. Mathematical detail is provided for the cases of turbulent flow over a rigid flat plate as well as a viscoelastic flat plate, where the invention accounts for both normal and longitudinal oscillations of the surface. A second object of the invention is the specification of material properties for a coating composed of multiple layers of isotropic viscoelastic materials. A third object of the invention is the specification of material properties for a coating composed of an anisotropic material. A fourth object of the invention is the minimization of edge effects for coatings of finite length. A fifth object of the invention is the stabilization of longitudinal vortices through combination of viscoelastic coating design with additional structure, such as riblets.
The methodology used herein to describe the interaction of a turbulent boundary layer (TBL) with a viscoelastic (VE) layer involves two tasks, 1) a fluids task, involving the calculation of turbulent boundary layer parameters, given boundary conditions for a rigid, elastic, or viscoelastic surface (herein referred to as the TBL problem), and 2) a materials task, involving the calculation of the response of a viscoelastic or elastic surface to a periodic forcing function which approximates the loading of the turbulent boundary layer. The invention focuses upon cation of amplitudes of surface oscillations and velocities, and of the energy flux for a viscoelastic coating (hereinafter referred to as the VE problem). These two tasks are coupled by coefficients related to surface boundary conditions of energy absorption and surface oscillation amplitudes (hereinafter referred to as dynamic and kinematic boundary conditions, respectively). The TBL problem is first solved for a rigid surface, thus providing necessary input to describe the forcing function on the surface, and also providing baseline calculations of friction drag, for comparison. The VE problem is solved next, given a periodic forcing function that approximates the shear and pressure pulsations of a given boundary layer. Initial choices for material parameters are based on theoretical and empirical guidelines. Optimal material parameters are chosen, following a series of iterations, such that the following two criteria are met:
1) the energy flux into the viscoelastic coating is maximum, and
2) the amplitudes of surface oscillation are less than the thickness of the viscous sublayer of the turbulent boundary layer over the coating.
If the amplitude of oscillations exceeds the thickness of the viscous sublayer of the turbulent flow, then the oscillations effectively increase the roughness of the surface, thus leading to an increase in friction drag. Furthermore, as the phase speed of disturbances in the boundary layer exceeds the shear wave speed in the material, a resonant interaction with large amplitude waves occurs. These conditions are to be avoided. Moderate energy flux into the material, however, where energy is transformed into internal shear waves and eventually dissipated as heat, leads to a qualitative and quantitative change in the turbulent energy balance, with a consequent reduction in friction drag. Optimal physical properties of a viscoelastic material will vary with freestream velocity, position along a body, pressure gradient, and any other factors which influence the development of the boundary layer and the characteristics of local turbulent fluctuations.
By solving the TBL equations, the turbulent friction drag over a viscoelastic, elastic, or rigid surface can be quantitatively evaluated. In the case of a viscoelastic surface, where energy is absorbed and surface oscillations are nonzero, both dynamic and kinematic boundary conditions are specified. These boundary conditions are derived directly from the solution of the VE equations for energy flux and surface oscillation amplitudes, and then transferred into a dissipation boundary condition and Reynolds stress boundary conditions for solution of the TBL equations. Vertical oscillations influence the effective roughness of the surface, and the root-mean square (rms) value of the vertical oscillation amplitude is classified as the dynamic roughness. If the oscillation amplitudes are lower than the viscous sublayer thickness, it is appropriate to estimate Reynolds stresses as zero. The equations for a turbulent boundary layer describe turbulent diffusion as a gradient approximation, which accommodates the dynamic boundary condition, and near-wall functions are introduced to describe for different surfaces the redistribution of turbulent energy in the near-wall region.